Consider the sphere of radius $r$ centered at 0 and the two great circles of the sphere lying on the xy and xz planes. A part of the sphere is shaved off in a such a manner that the cross section of the remaining part, perpendicular to the X-axis, is a square with vertices on the great circles. Compute the volume of the remaining part.
I'm not able to visualize even a bit of what's happening, in 3D. How is the cross section a square, and not a circle?
Also, as far as I know, a great circle is one that has the same center as its sphere.
Could someone please explain what's going on here, possibly with the help of figures/diagrams? All help is appreciated. Thanks a lot.
The "remaining part" looks like a rugby ball with 4 sides, with each side more flatten.
For each $x \in [-1,1]$, the cross section is the square/diamond formed by the points $(0, k), (k, 0), (0, -k), (-k, 0)$ in the $yz-$plane, where $k^2+x^2=1$.
Here's a 3D visualization, showing half of the solid. Some axes were swapped due to Google chart plotter's limitation. Here's a screenshot